The typical structure of dense claw-free graphs

Abstract

We analyze the asymptotic number and typical structure of claw-free graphs at constant edge densities. The first of our main results is a formula for the asymptotics of the logarithm of the number of claw-free graphs of edge density $γ\in (0,1)$. We show that the problem exhibits a second-order phase transition at edge density $γ^\ast=\frac{5-\sqrt{5}}{4}$. The asymptotic formula arises by solving a variational problem over graphons. For $γ\geqγ^\ast$ there is a unique optimal graphon, while for $γ<γ^\ast$ there is an infinite set of optimal graphons. By analyzing more detailed structure, we prove that for $γ<γ^\ast$, there is in fact a unique graphon $W$ such that almost all claw-free graphs at edge density $γ$ are close in cut metric to $W$. We also analyze the probability of claw-freeness in the Erdős-Rényi random graph $G(n,p)$ for constant $p$, obtaining a formula for the large-deviation rate function for claw-freeness. In this case, the problem exhibits a first-order phase transition at $p^\ast=\frac{3-\sqrt{5}}{2}$, separating distinct structural regimes. At the critical point $p^\ast$, the corresponding graphon variational problem has infinitely many solutions, and we again pinpoint a unique optimal graphon that describes the typical structure of $G(n,p^\ast)$ conditioned on being claw-free.